[Paper Review] Quasi-isometry classification of certain right-angled Coxeter groups

Pallavi Dani, Anne Thomas|arXiv (Cornell University)|Feb 25, 2014

Geometric and Algebraic Topology23 references3 citations

TL;DR

This paper classifies right-angled Coxeter groups WΓ with triangle-free, 1-ended defining graphs Γ up to quasi-isometry by analyzing splittings over 2-ended subgroups and using JSJ decompositions. It provides a complete quasi-isometry classification for hyperbolic WΓ when Γ is a generalized Θ graph, showing that some groups are quasi-isometric but not commensurable, extending prior commensurability results by Crisp–Paoluzzi.

ABSTRACT

Abstract. We investigate the quasi-isometry classification of the right-angled Coxeter groups WΓ which are 1-ended and have triangle-free defining graph Γ. We begin by char-acterising those WΓ which split over 2-ended subgroups, and those which are cocompact Fuchsian, in terms of properties of Γ. This allows us to apply a theorem of Papasoglu [21] to distinguish several quasi-isometry classes. We then carry out a complete quasi-isometry clas-sification of the hyperbolic WΓ with Γ a generalised Θ graph. For this we use Bowditch’s JSJ tree [8] and the quasi-isometries of “fattened trees ” introduced by Behrstock–Neumann [5]. Combined with a commensurability classification due to Crisp–Paoluzzi [11], it follows that there are right-angled Coxeter groups which are quasi-isometric but not commensurable. Finally, we generalise the work of Crisp–Paoluzzi [11]. 1.

Motivation & Objective

• To characterize right-angled Coxeter groups WΓ that split over 2-ended subgroups in terms of their defining graph Γ.
• To identify which WΓ are cocompact Fuchsian using graph-theoretic properties of Γ.
• To apply Papasoglu’s theorem to distinguish multiple quasi-isometry classes among hyperbolic WΓ.
• To achieve a complete quasi-isometry classification for hyperbolic WΓ when Γ is a generalized Θ graph.
• To generalize results of Crisp–Paoluzzi on commensurability and demonstrate quasi-isometric but non-commensurable right-angled Coxeter groups.

Proposed method

• Use graph-theoretic conditions on Γ to determine when WΓ splits over 2-ended subgroups.
• Apply Papasoglu’s theorem on quasi-isometry classification to distinguish classes based on splittings.
• Employ Bowditch’s JSJ tree decomposition to analyze the structure of hyperbolic WΓ with generalized Θ graphs.
• Utilize quasi-isometries of 'fattened trees' introduced by Behrstock–Neumann to compare geometric structures.
• Combine the geometric classification with Crisp–Paoluzzi’s commensurability results to compare quasi-isometry and commensurability classes.
• Generalize Crisp–Paoluzzi’s framework to extend the scope of known quasi-isometry and commensurability relationships.

Experimental results

Research questions

• RQ1 Which right-angled Coxeter groups WΓ with triangle-free, 1-ended Γ split over 2-ended subgroups, and how can this be characterized via properties of Γ?
• RQ2 What conditions on Γ make WΓ a cocompact Fuchsian group?
• RQ3 How can Bowditch’s JSJ tree and the theory of fattened trees be used to classify quasi-isometry types of hyperbolic WΓ when Γ is a generalized Θ graph?
• RQ4 Are there right-angled Coxeter groups that are quasi-isometric but not commensurable, and if so, how can this be demonstrated?
• RQ5 To what extent can the results of Crisp–Paoluzzi on commensurability be generalized in the context of quasi-isometry classification?

Key findings

• The paper provides a complete quasi-isometry classification for hyperbolic right-angled Coxeter groups WΓ when the defining graph Γ is a generalized Θ graph.
• It establishes that there exist right-angled Coxeter groups which are quasi-isometric but not commensurable, answering a key question in geometric group theory.
• The classification relies on a combination of JSJ decomposition and quasi-isometries of fattened trees, extending the applicability of geometric methods to this class of groups.
• The results generalize earlier work by Crisp–Paoluzzi, extending their commensurability classification to include quasi-isometry distinctions.
• The characterization of splittings over 2-ended subgroups and cocompact Fuchsian structures is fully determined by combinatorial properties of the defining graph Γ.
• The use of Papasoglu’s theorem enables the distinction of multiple quasi-isometry classes based on structural features of WΓ derived from Γ.